Direct simulation of non-premixed ﬂame extinction in a...

J. Fluid Mech. (2004), vol. 514, pp. 231–270. c© 2004 Cambridge University Press

DOI: 10.1017/S0022112004000266 Printed in the United Kingdom

231

Direct simulation of non-premixed flameextinction in a methane–air jet with

reduced chemistry

By C. PANTANOGraduate Aeronautical Laboratories, California Institute of Technology,

1200 E. California Blvd., 105-50, Pasadena, CA 91125, USA

[email protected]

(Received 10 November 2003 and in revised form 6 April 2004)

A three-dimensional direct numerical simulation (DNS) study of a spatially evolvingplanar turbulent reacting jet is reported. Combustion of methane with air is modelledusing a four-step reduced mechanism in the non-premixed regime. A total of eightchemical species are integrated in time along with the fluid mechanical fields. Thesolution of the compressible Navier–Stokes equations is obtained numerically formoderately low Mach number. A large computational grid, with 100 million gridpoints, is required in order to resolve the flame. The cold flow Reynolds number is3000. The focus of the study is to investigate the dynamics of extinction fronts in three-dimensional turbulent flows. A novel data reduction and identification algorithm wasdeveloped to postprocess the large DNS database and extract the shape of the evolvingflame surface including its edges and their propagation velocity. The joint probabilitydensity function (p.d.f.) of edge velocity and scalar dissipation was obtained andthe results indicate that the three-dimensional flame edges propagate with a velocitythat is largely controlled by the local rate of scalar dissipation, or equivalently interms of the local Damkohler number at the flame edge, as predicted by theory.Naturally, the effects of unsteadiness in this flow produce a broad joint p.d.f. Thestatistics collected also suggest that the mean value of the hydrogen radical reactionrate are very small in the turbulent regions of the flow owing to the functional formof the hydrogen radical reaction rate itself. The consequence of these results in thecontext of turbulent combustion modelling is discussed. Additional statistical andmorphological information of the flame is provided.

1. IntroductionExtinction dynamics in turbulent diffusion flames remains an open and challenging

subject. It is known that when a diffusion flame encounters a sufficiently large rate ofstrain (equivalent to the rate of scalar dissipation) the flame can extinguish owing toan imbalance of chemical heat production to diffusion from the flame (Peters 1986).The extinguished region and the burning flame are separated by a flame edge with astrain-rate-dependent structure (Vervisch & Poinsot 1998). This extinction front canexpand or collapse depending on the dynamics of the flow, transport properties andthe chemistry details. Flame surface can also be produced under certain conditionsthat are generally referred to as turbulent reignition. Two mechanisms are thoughtto be responsible for flame creation. In one mechanism, flame edges that separate theburning from the quenched regions, propagate against the flow and they are able to

232 C. Pantano

heal or close the extinguished region. In the other mechanism, reignition can take placewhen hot pockets of reacted products and mixed non-burning reactants are broughtinto contact, through convection and diffusion, for a sufficient amount of time. Hotpockets of products are left-overs from the pre-quenched state or can be convectedfrom nearby burning flame sections. Mixing of reactants takes place through diffusionin the post-quenched region. In jet diffusion flames close to extinction, this mechanismmay not play a significant role in the near-field region of the jet, where the rate of strainis largest, if both streams are cold because the ignition delay time of typical hydrocar-bon mixtures at low temperatures is too large in comparison with the characteristicflow time. Only the first mechanism seems dominant. On the other hand, furtherdownstream of the jet, the level of turbulence may be sufficiently strong to quench theflame locally and reignition through the second mechanism could take place. These twomechanisms for flame creation together with the extinction mechanism due to largerate of strain are considered to be dominant in flame extinction reignition dynamics.

The study of edge-flame dynamics is relatively well advanced, at least in one- andtwo-dimensional configurations (including axisymmetric). The first experimentalevidence is due to Phillips (1965) and theoretical results date back to Linan &Crespo (1976), Dold (1988) and Buckmaster & Matalon (1988). Buckmaster (2002)reviewed the current understanding of the dynamics of flame edges. In general, theflame edges are composed of two premixed branches, a rich and a lean branch, anda diffusion flame aligned with the stoichiometric line in what is called colloquiallya triple flame. For large values of the strain rate, the two premixed branches mergeinto a single edge. In all these cases, there is a well-defined edge propagation velocity,referred to here as the edge-flame velocity, that depends on the Damkohler number(the flow to chemistry time scale ratio), the Lewis number (the thermal to moleculardiffusivity ratio) (Buckmaster 1996, 2001) and the level of heat release. This velocitycan be negative if the rate of strain is sufficiently large. Theoretical descriptionsof triple edge-flames using the large activation energy asymptotic approximationwith zero heat release (Daou & Linan 1998) and with finite heat release (Ghosal& Vervisch 2000) have been developed. Detailed numerical studies have also beencarried out for freely propagating edge-flames without the effects of heat releaseby Kioni et al. (1993) and with the effects of heat release by Ruetsch, Vervisch& Linan (1995) and Echekki & Chen (1998). In the interest of understanding theinteraction of the flame edges in more complex flows, some studies have consideredthe interaction of the edge-flame with a counterflow that is perpendicular to the planeof the flame, also called a strained mixing layer (Daou & Linan 1998; Vedarajan& Buckmaster 1998; Buckmaster & Short 1999; Thatcher & Dold 2000; Short,Buckmaster & Kochevets 2001). Experiments have also been performed. Shay &Ronney (1998) studied the effects of variable strain rate in space and showed theformation of stable edge-flames. In the case of triple-edge flames, Ko & Chung(1999) performed experiments with methane–air jets and report that their unsteadyedge flames propagate at a speed that increases with decreasing mixture fractiongradient, in agreement with theoretical predictions. Santoro, Linan & Gomez (2000)have performed experimental measurements of methane–air flames in a counterflowmixing layer and find the existence of standing edge-flames, with triple-flames forlarge Damkohler number and simple edge-flames for lower Damkohler numbers.

In the case of three-dimensional turbulent flows, most studies have been carried outwith the aid of experimental diagnostic techniques (Everest et al. 1995; Kelman &Masri 1997; Muniz & Mungal 1997, 2001; Starner et al. 1997; Barlow & Frank 1998;Rehm & Clemens 1999; Meier et al. 2000; Dally, Karpetis & Barlow 2002; Karpetis &

Direct simulation of methane–air flame extinction 233

Barlow 2002). Several studies have documented the formation of extinction pockets indiffusion flames, see Everest et al. (1995) and Kelman & Masri (1997) among others.The most detailed experimental data corresponds to planar cuts of the physicaldomain of interest and the three-dimensional structure of the flame is not examined.For this reason, numerical simulation can aid in the pursuit of a better understandingof extinction dynamics. Unfortunately, the computational turbulent combustion com-munity faces a large disparity between available computational resources and the re-quirements of fully turbulent reacting flows involving realistic chemistry. The majorityof detailed chemistry simulations are only accessible in two dimensions, even in thelargest supercomputers. Only recently has numerical simulation become sufficientlypowerful to attack three-dimensional flows. Here, we understand numerical simula-tions as direct numerical simulation (DNS) in which all temporal and spatial scales ofthe flow and the chosen chemistry are accurately resolved. In their review of DNS ofnon-premixed turbulent combustion Vervisch & Poinsot (1998) identified four differenttypes of relevant analysis. The first three types of analysis identified by Vervisch &Poinsot (1998) have been investigated in the past, see Pantano, Sarkar & Williams(2003 and references therein) for the case of typical heat release in a methane–air shearlayer. The present paper is centred around the fourth type of analysis, concerningeffects of finite-rate chemistry. In the present study, we concentrate on scalar fieldsthat are active, that is, they affect pressure, density or velocity fields. The couplingtakes place through variations of the density owing to heat release and owing to thepresence of non-zero chemical source terms of finite or infinite rate (reactive fields).DNS of active reactive scalars has been discussed in reviews (Jou & Riley 1989;Givi 1989; Vervisch & Poinsot 1998). The flow configurations considered range fromhomogeneous isotropic turbulence (Mell et al. 1994; Mahalingam, Chen & Vervisch1995; Swaminathan, Mahalingam & Kerr 1996; Montgomery, Kosaly & Riley 1997;Swaminathan & Bilger 1997; Bedat, Egolfopoulos & Poinsot 1999; Livescu, Jaberi &Madnia 2002), temporally evolving turbulent shear layers (McMurtry, Riley &Metcalfe 1989; Miller, Madnia & Givi 1994; Pantano et al. 2003), spatially evolvinggrid turbulence (Cook & Riley 1996) and jets (Mizobuchi et al. 2002). Of all theseworks, that of Mizobuchi et al. (2002) is the most relevant in our context. Theyperformed a simulation of a three-dimensional lifted hydrogen flame issuing from asquare duct and used a detailed chemical mechanism of hydrogen–oxygen combustion.In their case, owing to the very short reaction times characteristic of hydrogen com-bustion, no flame holes were observed.

With our current computational resources, the most promising chemistry modelsthat can be incorporated in three-dimensional simulations of turbulent combustionare restricted to reduced chemical mechanisms. Past works include Swaminathan &Bilger (1997), who investigated a model two-step chemical mechanism for methane–aircombustion and Montgomery et al. (1997) who used a three-step reduced mechanismto simulate hydrogen–oxygen non-premixed combustion. Bedat et al. (1999) used anintegrated combustion chemistry (ICC) methodology in which the chemical scheme ispostulated and the parameters of the scheme are determined by matching several flameproperties. In the present study, we are interested in methane–air combustion and wechose a chemical mechanism that is sufficiently complex to include as many details ofthe chemical structure of the flame as possible while being computationally tractable.This mechanism is the four-step reduced mechanism of Peters (1985) and was selectedfor several reasons. First, it is shown by Peters (1985) that the mechanism is deducedsystematically from a skeletal C-1 mechanism assuming steady-state approximationsof some radicals and partial equilibrium for some reactions. Thus, there is some

234 C. Pantano

degree of connection with the more complete chemistry (the mechanism is not adhoc). Secondly, it has been shown by Peters (1985) that predictions of extinction strainrate resulting from this reduced mechanism are in good agreement with those obtainedusing the full mechanism and that the internal structure of the flame is reproducedwell both qualitatively and quantitatively for most species. Thirdly, extensive studiesof the asymptotic structure of the flame using the reduced mechanism are available(Seshadri & Peters 1988; Bai & Seshadri 1999). In these studies, rate-ratio asymptoticsis used to understand the internal structure of the flame. It is known that the flameis composed of the classical external Burke–Schumann structure. The inner structureis composed of a thin H2 − CO oxidation layer of thickness O(ε) towards the leanside, a thin water gas shift non-equilibrium reaction of thickness O(ν) and a thinfuel consumption layer towards the fuel side of thickness O(δ). Analysis of the innerstructure for large values of the Damkohler number shows that ε > ν > δ. There iseven some work on the inner structure of methane–oxygen–nitrogen diffusion flames(Chelliah & Williams 1990). Lastly, the mechanism has been successfully used byseveral authors in one- and two-dimensional flows (see Peters & Kee 1987; Card et al.1994; James & Jaberi 2000).

The objective of the present study is to investigate the behaviour of flame edgeswith complex chemistry in three-dimensional flows. The present work addressesrealistic heat release and requires the computations of 8 scalar species employing asmany as 100 million grid points. A fully compressible code, similar to that used inPantano et al. (2003), is employed, with a convective Mach number (Bogdanoff 1983;Papamoschou & Roshko 1988), defined as Mc = �u/(c1 + c2) (where �u is the velocitydifference between the coflow and the jet, and c1 and c2 are the speeds of sound ofeach stream), equal to 0.3. This value is small enough that compressibility effectsfrom Mach number are not important (Pantano et al. 2003). For future reference, weintroduce the concept of mixture fraction and scalar dissipation. A common approachin the modelling of non-premixed turbulent combustion is based on knowledge oftwo variables; a mixture fraction, Z, that represents the mixture composition, givingthe fraction of the material that comes from the fuel stream, and its so-called scalardissipation, χ = 2D∇Z · ∇Z (in which D is its molecular diffusivity), χ being relatedto the rate of dissipation of fluctuations of Z in turbulent flow (see Williams 1985).These two quantities are used in the analysis of the results described in § 6.

2. The flow configurationFigure 1 is a sketch of the planar jet (a model of a slot burner) considered here. The

jet velocity is Uj , the coflow velocity is Uc and the velocity difference is �u = Uj − Uc.The domain size is L1 in the streamwise direction, L2 in the transverse direction andL3 in the spanwise direction. The jet height is denoted by H . The jet is composedof a mixture of methane and nitrogen. The coflow is composed of air approximatedas a mixture of oxygen and nitrogen. Both, methane and oxygen, mass fractions areequal to 0.23. These values were chosen because the global chemistry that occurs atthe flame,

CH4 + 2O2 → CO2 + 2H2O, (2.1)

then yields a stoichiometric mixture fraction Zs = 0.2. A more complete argumentregarding our choice of stoichiometry can be found in Pantano et al. (2003).

In order to reduce the computational cost associated with full chemistry models,the reduced mechanism of Peters (1985) for combustion of methane was chosen in


x2

x1

L2

Up

∆u

L1

H

h

Uj

Uc

Figure 1. Schematic diagram of the spatially evolving jet with streamwise velocity profileparameters shown in a two-dimensional projection (x = x1, y = x2, z = x3).

this study. In this four-step mechanism, derived from a skeletal C-1 mechanism bysystematic application of quasi-steady state and partial equilibrium approximations,the resulting non-linear relationships between the mass fractions of the species inquasi-steady state are truncated. This renders the algebraic expressions of the reactionrates explicit. The mechanism involves N = 8 species, namely, CH4, O2, H2O, CO2,CO, H2, H and N2. Thus, seven transport equations with non-zero reaction rates mustbe solved along with the flow variables. The mass fraction of N2 is obtained fromthe balance of all species and no transport equation is thus required for this inertspecies. The reaction-rate expressions for the rates of production–consumption ofspecies are provided as algebraic expressions of the concentrations and temperaturein Seshadri & Peters (1988).

To make the influence of density variation exclusively associated with heat release,the jet and coflow have the same density. To also make their pressures equal requires atemperature ratio equal to the average molecular weight ratio of the air in the coflowto that of the fuel in the jet (ideal gas at low Mach number). Thus, the air tempera-ture is 20% higher than the fuel temperature of 298 K. Specific heats of the speciesin the ideal gas mixture are allowed to depend on temperature, to maintain correctcold-gas values and avoid achieving flame temperatures that are too high at thereaction sheet, which would result in unrealistically low gas densities. The specificheats at constant pressure and enthalpy were obtained from NASA polynomial fits(McBride, Gordon & Reno 1993). The values of these parameters give an adiabaticflame temperature for Zs = 0.2 of Tf = 2022 K.

To clarify interpretations by focusing attention on as few different physical pheno-mena as possible, simplifications were introduced in molecular transport properties.The viscosity µ was taken to be proportional to T m. All chemical species were assumedto have diffusion coefficient, Di , that have the same temperature dependence, namely,

236 C. Pantano

CH4 O2 H2O CO2 CO H2 H N2

0.97 1.11 0.83 1.39 1.10 0.30 0.18 1.00

Table 1. Constant Lewis number of involved species used in the simulation(Smooke & Giovangigli 1991).

ρDi is proportional to T m so that the Schmidt number, Sci = µ/ρDi , is constant. Alsoimposed is constancy of the Prandtl number, Pr= µCp/κ , where κ denotes the thermalconductivity. Because of the variations of the specific heat Cp of the mixture, κ alsovaries to maintain Pr constant. The approximate value for air, Pr = 0.7, is employedthroughout. The reference values of Di at To were obtained from Smooke &Giovangigli (1991) and were such that the Schmidt number is constant and equalto the product of the Prandtl number times the Lewis number, Sci = Pr Lei , whereLei = κ/(CpρDi). The values of the Lewis numbers of the different species are specifiedin table 1. The effects of differential diffusion are thereby taken into account in thissimplified transport model.

To enhance flame stability at the inflow and avoid flame lift-off or blow-out, apilot is inserted between the jet core and the main coflow. This pilot is implementednumerically as a thin coflow with a high temperature, equal to the adiabatic flametemperature of the jet–coflow mixture stoichiometry. Moreover, the pilot streamwisevelocity is slightly higher than that of the main coflow to avoid recirculation. Thistechnique has been used by Wall, Boersma & Moin (2000) to stabilize round jet flames.The flame at this pilot conditions burns below the quenching scalar rate of dissipationand remains attached to the inflow of the domain.

3. FormulationThe unsteady three-dimensional compressible Navier–Stokes equations for a

Newtonian fluid composed of a reacting ideal-gas mixture are considered in thisstudy. Energy conservation is written as a pressure equation to facilitate computation.Relevant parameters are the Reynolds number,

Re =ρo�uH

µo

, (3.1)

the non-dimensional heat release,

Q =qoYF,f Zs

CpNoToνF WF

, (3.2)

and the Damkohler number,

Da =toWOAo

ρo

. (3.3)

In (3.1), µo is the viscosity of the mixture at To and in (3.2), qo denotes the enthalpyof the reaction, (2.1), Williams (1985),

qo =

N∑i=1

νiWi�hoi . (3.4)

The enthalpy of formation of species i is denoted by �hoi , Wi is the molecular weight

of species i, νCH4= νF = 1, νO2

= νO = 2, νCO2= −1, νH2O = −2 and CpNo

is the specific


heat of nitrogen at To. The molecular weights, Wi , are dimensional quantities in thispaper (units of gram per mol). The reference molecular weight is that of oxygen, O2,and is denoted by WO . In (3.3), Ao is a characteristic reaction rate (units of molesper unit volume and time) and is specified below in terms of one of the reactionrates of the chemical mechanism. The choice of characteristic chemical time is notunique for a multistep mechanism and it is discussed in § 3.2. The formulation isnon-dimensional, the unit length being H , velocity �u, time to =H/�u, density ρo,temperature To, enthalpy CpOo

To and pressure ρo�u2. The inert mass fraction, N2, isdetermined from

YN2= 1 −

∑i �=N2

Yi. (3.5)

Here, the subscripts O and F stand for oxidizer, O2, and fuel, CH4, respectively, andYO,o is the mass fraction of oxygen in the oxidizer (air) stream, while YF,f is the massfraction of fuel (methane) in the fuel stream. The stoichiometric mixture fraction, Zs ,is equal to

Zs =1

φ + 1, (3.6)

where φ = (WOνOYF,f )/(WF νF YO,o) is the fuel–air equivalence ratio. The Mach numberis M = �u/

√γoRoTo, γo and Ro being the ratio of specific heats and gas constant for

O2 at To, and the normalized average molecular weight is

W =

(WO

N∑i=1

Yi

Wi

)−1

. (3.7)

3.1. Governing equations

The conservation equation for species mass fractions, Yi , is

∂(ρYi)

∂t+

∂(ρYiuk)

∂xk

=1

Re Sci

∂

∂xk

(δ∗ ∂Yi

∂xk

)+ Da ωi, (3.8)

where the reaction rate term ωi is given in § 3.2 for each species. The conservationequations for mass, momentum and energy are

∂ρ

∂t+

∂(ρuk)

∂xk

= 0, (3.9)

∂(ρui)

∂t+

∂(ρukui)

∂xk

= − ∂p

∂xi

+∂σik

∂xk

, (3.10)

and

∂p

∂t+ uk

∂p

∂xk

= −γp∂uk

∂xk

+(γ − 1)

(γo − 1)Re PrM2

∂

∂xk

(κ∗Cp

∂T

∂xk

)

+ (γ − 1)Φ +(γ − 1)

(γo − 1)ReM2

N−1∑i=1

Cpi − CpN

Sci

δ∗ ∂T

∂xk

∂Yi

∂xk

+γ T

γoReM2

N−1∑i=1

(1

Wi

− 1

WN

)WO

Sci

∂

∂xk

(δ∗ ∂Yi

∂xk

)

+Da

M2

N−1∑i=1

(γ

γo

WO

Wi

T − γ − 1

γo − 1hi

)ωi . (3.11)

238 C. Pantano

In (3.10), the viscous stress tensor is given by

σij =µ∗

Re

{∂ui

∂xj

+∂uj

∂xi

− 2

3

∂uk

∂xk

δij

}, (3.12)

and in (3.11), the viscous dissipation is

Φ = σij

∂ui

∂xj

. (3.13)

The non-dimensional average specific heat of the mixture is

Cp =

N∑i=1

Cpi(T )Yi, (3.14)

where the specific heats at constant pressure, Cpi(T ) are expressed as polynomialfunctions of the temperature with coefficients given by McBride et al. (1993). Theenthalpy, hi , is defined by

hi =�ho

i

CpOoTo

+

∫ T

1

Cpi(T ) dT . (3.15)

The non-dimensional equation of state of the mixture is

p =ρT

γoM2W. (3.16)

The specific heat ratio of the mixture, γ , varies somewhat and is given by

γ =γo

γo − (γo − 1)/WCp

. (3.17)

The non-dimensional transport coefficients µ∗, δ∗ and κ∗ are given by

µ∗ = δ∗ = κ∗ = T m, (3.18)

with m = 0.7. The heat-release parameter Q of (3.2) is equal to 7.45 and it would beequal to Tf /To − 1 if the specific heat of the mixture were constant.

Finally, a mixture fraction field, Z, is computed along with the rest of the variables.This field obeys the following transport equation,

∂(ρZ)

∂t+

∂(ρZuk)

∂xk

=1

Re Sc

∂

∂xk

(δ∗ ∂Z

∂xk

), (3.19)

where the mixture fraction Schmidt number is Sc=Pr. This implies that the Lewisnumber is one for this field. The Z field is used to initialize the flame and to help inthe interpretation and extraction of statistical information.

3.2. Chemistry model

Peters (1985) reduced mechanism can be represented by the following global reactions

CH4 + 2H + H2O = CO + 4H2 (I ),

CO + H2O = CO2 + H2 (II ),

H + H + M = H2 + M (III ),

O2 + 3H2 = 2H + 2H2O (IV ),


Reaction Ai βi Ei

k1 1.2 × 1017 −0.91 69.10k5 2.0 × 1018 −0.8 0.0k10 1.656 × 107 1.5247 60.042k11 2.2 × 104 3.0 36.6

Table 2. Specific dimensional reaction-rate parameters. Units in cm, mole, Kelvin and kJ.

with corresponding non-dimensional reaction rates given by

ωI = k11CCH4CH, (3.20)

ωII = k10

(CH/CH2

)(CCOCH2O − CCO2

CH2/KII

), (3.21)

ωIII = k5CO2CHCM, (3.22)

ωIV = k1CH

(CO2

− C2HC2

H2O/C3

H2KIV

). (3.23)

The non-dimensional concentrations, Ci , are defined as

Ci =ρYiWO

Wi

, (3.24)

and the third body concentration, CM , is defined as

CM =

N∑i=1

ηiCi, (3.25)

with catalytic efficiencies ηCH4= ηH2O =6.5, ηO2

= ηN2= 0.4, ηCO2

= 1.5, ηCO = 0.75 andηH2

= ηH = 1 (Smooke & Giovangigli 1991). The mechanism given by (I)–(IV) is aglobal representation of the chemistry and should not be confused with the actualpaths that the reaction takes. These are not elementary reactions; their rates areexpressed as algebraic functions of rates appearing in the skeletal C-1 mechanism.These reaction-rate constants are given in the customary Arrhenius form,

ki = AiTβi e−Ti/T , (3.26)

where Ti = Ei/R with R the universal gas constant, equal to 8.314 Jmol−1 K−1 and Ei

is the activation energy of the elementary reaction i. In (3.20) to (3.26), all parametersare non-dimensional. The remaining constants, KII and KIV, are

KII = 3.9512 10−3 T 0.8139 e16.6247/T , (3.27)

KIV = 2.7405 T −0.2484 e19.262/T . (3.28)

The values of the parameters appearing in (3.26) were obtained by non-dimension-alizing the dimensional rate constants reported in Seshadri & Peters (1988) andshown in table 2 by the largest of the rates at To, in this case, that of reaction k5. Thisdimensional rate is given by

Ao = A5Tβ5o e−E5/RTo

(ρo

WO

)3

, (3.29)

and was used in (3.3) to define the Damkohler number.

240 C. Pantano

The reaction rates, ωi , appearing in (3.8) are defined in terms of (3.20)–(3.23) by

ωCH4= −WCH4

WO

ωI , (3.30)

ωO2= −WO2

WO

ωIV, (3.31)

ωH2O =WH2O

WO

(2ωIV − ωII − ωI ), (3.32)

ωCO2=

WCO2

WO

ωII, (3.33)

ωCO =WCO

WO

(ωI − ωII), (3.34)

ωH2=

WH2

WO

(4ωI + ωII + ωIII − 3ωIV), (3.35)

ωH = 2WH

WO

(ωIV − ωI − ωIII). (3.36)

A common difficulty encountered in the implementation of reduced mechanisms,as the one considered here, is the presence of algebraic terms in the denominator ofthe reaction-rate expressions. When the denominator goes to zero, the reaction ratebecomes infinitely large. In our case, the presence of the concentration of H2 in thedenominator of (3.21) and (3.23) leads to this undesired behaviour. In regions wherethere is no H2, these expressions diverge to infinity and an appropriate regularizationmust be applied for numerical purposes. As suggested by Peters (1991), a commonregularization is to add a small constant, εo, to the denominator of (3.21) and (3.23) sothat 1/CH2

becomes 1/(CH2+εo), in order to avoid the singularity. This regularization

was sufficient in a one-dimensional flamelet test calculation. Unfortunately, in oursimulation, it was found that shifting the hydrogen concentration by εo was notsatisfactory at all points of the domain. We could still find very compact regionswith unphysically high values of the reaction rates far away from the flame. Aftersome trial and error, it was decided to regularize the algebraic singularity in hydrogenconcentration with

1

CH2

→

0, 0 � CH2< εo,

tanh

(CH2

− εo

εo

)1

3εo

, εo � CH2< 3εo,

1

CH2

, 3εo � CH2.

(3.37)

Here, the value of εo was chosen to be approximately equal to 1% of the maximumconcentration of H2 in the simulation, namely, εo = 1.6 × 10−5. These choices alloweda very smooth transition of the reaction rates from the flame region to the regionswhere CH2

was zero.

3.3. Numerical scheme, flow initialization and boundary conditions

The simulation proceeds in the following way: suppose that the variables ρ, p, ui andYi are available at a given time. The temperature is obtained from (3.16) after using(3.7). The enthalpy and specific heat are then computed from (3.14)–(3.15). Finally,


(3.8)–(3.11) are solved using a semi-implicit time integration to advance the variablesin time.

In direct simulation of multi-species turbulent reactive flows close to flame extinc-tion, the flow and chemical time scales are comparable, but the spatial resolutionrequired for chemistry demands fine grids in order to resolve the thin reaction zones.Furthermore, some of the chemical species can be very diffusive, in our case, thesespecies are hydrogen and hydrogen radical (see table 1). The time step allowed whenintegrating the governing equations with explicit schemes is controlled by the diffusivestability limit dictated by these few chemical species. We overcome this problemby integrating implicitly the diffusive terms of the hydrogen and hydrogen radicalgoverning equations, while all other terms are integrated explicitly. In our case, weuse a third-order additive semi-implicit Runge–Kutta scheme (Pantano 2004).

Spatial derivatives are computed using a compact Pade scheme in space of sixth-order of accuracy (Lele 1992). Characteristic inflow boundary conditions are imposedin the streamwise direction, x1, and ‘non-reflective’ boundary conditions are imposedin the x2-direction (Baum, Poinsot & Thevenin 1995; Stanley, Sarkar & Mellado 2002).The grid was uniform in the x1- and x3-directions with an equal grid spacing, �x, inboth directions. In the transverse direction, x2, the grid is uniform across the centreof the domain enclosing the thickness of the jet and it is stretched gradually in therest of the domain. The grid spacing in the centre of the domain in the transversedirection is also �x, while the stretching was 1% in the corresponding part of thedomain.

The flow is initialized to a hyperbolic-tangent profile for the mean streamwisevelocity, u1(x2),

u1(x2) =

Uj + Up

2+

Uj − Up

2tanh

(− (x2 − H/2)

2δo

), x2 <

H + h

2,

Up + Uc

2+

Up − Uc

2tanh

(− (x2 − H/2 − h/2)

2δo

), x2 >

H + h

2,

(3.38)

while the transverse mean velocity components are set to zero. The symmetric partof the jet is initialized by mirroring the solution with respect to the symmetry axis.The coflow velocity is denoted by Uc, the jet velocity by Uj = Uc + �u and the pilotvelocity is denoted by Up . The value of δo =0.05H is employed in the simulation.The mean pressure is set initially to a uniform value and ρj = ρc = 1 throughout,where ρj is the jet density and ρc is the density of the coflow stream. In additionto the mean fields, broadband fluctuations are used to accelerate the transition toturbulence. This is achieved by generating a random velocity field with an isotropicturbulence spectrum of the form

E(k) = (k/ko)4 exp (−2(k/ko)

2), (3.39)

where k is the wavenumber and ko the wavenumber of peak energy. The extent of theinitial velocity fluctuations is limited in the cross-stream direction by an exponentialdecay given by,

exp(−((x2 ± H/2)/δb)2), (3.40)

where δb = δo. The initial pressure fluctuations are obtained from the Poisson equationfor incompressible flow.

242 C. Pantano

The species mass fraction were initialized in a two-stage process. First, a passivescalar, Z, representing a mixture fraction variable was initialized to

Z(x2) =

1 + Zs

2+

1 − Zs

2tanh

(− (x2 − H/2)

2δo

), x2 <

H + h

2,

Zs

2+

Zs

2tanh

(− (x2 − H/2 − h/2)

2δo

), x2 >

H + h

2,

(3.41)

with initial scalar fluctuations set to zero. The temperature and density were setto the Burke–Schumann values, T e(Z) and ρe(Z) (Williams 1985), respectively. Thesimulation was run for a number of time steps, of the order of two flow transienttimes, based on the jet exit velocity, Lx/Uj , until the jet instability modes developand the unphysical initial conditions are washed out of the domain. Secondly, aone-dimensional flamelet calculation was carried out to obtain Y e

i (Z). The flameletsolution was obtained from the steady flamelet equation of Peters (1984),

− ρχ

2Lei

d2Y ei

dZ2= Da ωi(Y

e, T ), (3.42)

with the scalar dissipation given by χ = 8Zs/Reδ2o . Solution of the boundary-value

problem, (3.42), gives the mass fraction of all species and temperature as a function ofZ. The species mass fractions were then initialized through the mapping Yi = Y e

i (Z),where Z was the result of the previous initialization step. Prescription of the initialscalar field gives initial distributions of Yi and W from previous equations.

The computational domain is composed of two parts. A so-called inflow domainthat contains streamwise periodic flow that is convected into the primary largerdomain. This technique is described, for example, by Li, Balaras & Piomelli (2000)and by Stanley et al. (2002) for planar spatially evolving jets. The data contained inthe inflow domain are obtained by performing a temporal simulation (with periodicstreamwise boundary conditions) for a short but otherwise sufficient time to allow thedesired level of inflow fluctuations to be injected in the primary domain. The temporalsimulation data at one instant in time (frozen flow) is then convected at constantspeed, (Uj + Uc)/2, using Taylor’s hypothesis to relate spatial to temporal derivatives;required by the incoming characteristic boundary conditions of the spatial simulation.The peak turbulence intensity level of the inflow forcing is around 4%.

Since the kinematic viscosity increases with temperature, the Reynolds number, Re,was deliberately kept large at 3000. The Mach number that appears in (3.11) was setto M =0.694 and the Damkohler number was set to Da = 5000. The composition ofthe coflow and the jet is YO,o =0.23 and YF,f =0.23 and gives a stoichiometric mixturefraction value of Zs = 0.2. The Prandtl number is 0.7. The main coflow velocity, Uc/�u

is 0.03, the pilot velocity, Up/�u is 0.3 and the pilot width, h/H , is 0.325. If we assumeatmospheric pressure and the viscosity of air at 298 K, H is approximately equal to2mm for the values of Reynolds and Damkohler numbers of this simulation. This jetheight is similar to the jet diameter used by Mizobuchi et al. (2002) in a simulation ofa hydrogen–air lifted jet. The number of grid points was Nx = 1024 in the streamwisedirection, Ny = 512 in the transverse direction and Nz = 192 in the spanwise direction.The total number of grid points is roughly 100 million and there were 13 variablesthat had to be integrated, five fluid mechanical and 8 scalars. The large resolutionrequirements were mandated by the need to resolve the fuel consumption layer of themethane–air mechanism (Seshadri & Peters 1988). It is well known that this regionmust be well resolved in order to avoid numerical extinction of the flame owing to lack


1x

0

0.05

0.10

0.15

0.20

0.2 0.4 0.6 0.8 1.0z

0

0.05

0.10

0.15

0.20

0.25

2 3 4 5

(a) (b)

Figure 2. Mass fraction profiles: (a) premixed flame as a function of distance and (b) flameletsolution close to extinction as a function of mixture fraction. �, CH4; �, O2; �, H2O;�, CO2; �, CO; �, H2.

of resolution. In our case, a resolution of approximately 10 points across the fuel con-sumption layer was found to be sufficient with our numerical scheme (Vervisch &Poinsot 1998). The analogous flow with single-step or infinitely fast chemistry cantypically be resolved with at least half the resolution in each direction. For a three-dimensional flow, this implies a cost reduction in space of approximately an orderof magnitude. As pointed out in Swaminathan & Bilger (1997), the simplificationsinvolved in deriving the reduced mechanism of Peters lead to an excessively thin fuelconsumption zone, while maintaining good extinction characteristics. It is possibleto artificially alter the rates of the fuel consumption zone and make the resolutionrequirements less demanding, but in the present study the original mechanism was usedwithout modifications. The simulation was run for approximately two transient times.It required 340 000 processor hours of the ASCI QSC system at Los Alamos NationalLaboratory. The simulations used either 128 or 256 processors depending on theavailability of the queuing system and took approximately four months to complete.

For future reference in the analysis of the flame edge results, it is necessary to obtainsome additional one-dimensional flame values that are useful in the discussion of theresults. Figure 2 shows the mass fractions of CH4, O2, H2O, CO2 and CO in a one-dimensional premixed planar flame (figure 2a) and a flamelet solution of (3.42) closeto extinction (figure 2b). The premixed flame solution was obtained using the reducedmechanism with the compressible formulation and the values of the parameterspreviously discussed. The premixed planar flame was computed at the compositioncorresponding to the frozen flow mixture with mixture fraction equal to Zs . For unityLewis number, this is the appropriate mixture composition encountered ahead of theflame-edge head (Daou & Linan 1998). We denote the premixed planar flame speedvalue as SL,st and numerical integration gives the non-dimensional value of 0.022.The resolution used in the calculation of the premixed planar flame was identical tothat of the three-dimensional simulation and corresponds to 12 grid points for thehydrogen radical reaction rate. This resolution was found to be numerically appro-priate. Finally, the diffusion flamelet structure in figure 2(b), was computed very closeto the extinction limit, where the non-dimensional quenching scalar dissipation, χq ,had a value approximately equal to 0.0205.

244 C. Pantano

10

8

6

4

2

0 5 10 15x

y

7.0

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

Figure 3. Temperature isocontours at plane through the centre of the domainat an instant in time.

4. Qualitative description of the flameTurbulent flames, including the flame considered in this study, are complex three-

dimensional objects that change in time owing to the unstable nature of the flow and tothe effects of extinction and reignition. Figure 3 shows isocontours of temperature atthe centre of the domain midway through the simulation. This figure is representativeof the behaviour of the temperature field at other times. The parameters of thissimulation were chosen to produce a flame that is partially extinguished and, in ourcase, this occurs predominantly around the centre of the domain. In this region,the large-scale organized vortices that are shed from the shear layers have sufficientstrength, large rate of strain, to extinguish the flame. This is observed in figure 3, wherelow temperature values can be seen around the centre of the figure at both edgesof the jet. The presence of these vortical structures that promote extinction in ourflame are commonly seen in the near-field region of turbulent diffusion flames. Here,quasi-laminar flame structures envelop the jet core where vortical structures exist. Thisbehaviour is observed in experimental and numerical observations of diffusion flames(Yule et al. 1980; Chen et al. 1991; Schefer et al. 1994; Everest, Feikema & Driscoll1996; Takahashi et al. 1996; Yamashita, Shimada & Takeno 1996). Using a reactiveMie scattering technique, Roquemore et al. (1987) showed the existence of these vor-tical structures entrapped within the jet core edges and surrounded by the flame in thenear-field region. In a comparative study by Clemens & Paul (1995) where both non-premixed reacting and non-reacting jets were analysed, it was also found that the nearfield consisted of laminar regions surrounding the inner core, where organized vorticalstructures were visible. Comparing experimental results for reacting and non-reactingjets, they show that the strong density gradients induced by combustion are responsibleof extending the potential core. Large-scale organized vortical structures are also seenin the experimental measurements of CH4/H2/N2 flames by Bergmann et al. (1998).Laser induced fluorescence intensity of NO shows the presence of these structures atthe edges of the jet core, which is formed by the shear layers separating the fuel fromthe oxidizer. In our piloted flame, the region close to the inflow where the pilot is still


10

8

6

4

2

0 5 10 15

y

10

8

6

4

2

0 5 10 15

10

8

6

4

2

0 5 10 15

y

10

8

6

4

2

0 5 10 15

10

8

6

4

2

0 5 10 15

y

10

8

6

4

2

0 5 10 15

0.020.040.060.080.100.120.140.160.180.200.22

0.010.020.030.040.050.060.070.080.09

0.00020.00040.00060.00080.00100.00120.00140.00160.00180.00200.00220.00240.0026

2.04.06.08.0 × 10–51.01.21.41.61.82.02.2 2.4 2.62.8 × 10–4

0.0080.0100.0150.0200.0250.0300.0350.0400.0450.050

0.020.040.060.080.100.120.140.160.180.200.22

(a) (b)

(c) (d)

(e) ( f )

x x

Figure 4. Species mass fraction isocontours at a plane through the centre of the domain atan instant in time. (a) Methane, (b) oxygen, (c) water, (d) carbon monoxide, (e) molecularhydrogen and (f ) hydrogen radical.

strong is approximately laminar. However, the extent of this region is reduced herebecause the inflow forcing we use is strong. We chose this level of forcing to reduce theextent of the potential core so that the usefulness of the computational domain is max-imized to capture more turbulent flow within the box. As an aside, it has been shownthat the region close to the nozzle experiences stronger differential diffusion effects(non-unity Lewis number effects) owing to the quasi-laminar behaviour of the flame(Bergmann et al. 1998). In round jets, this region extends from the nozzle to x/D ≈ 10,where D is the diameter of the jet and the effects of differential diffusion remain evenat larger distances from the nozzle (Pitsch 2000). In our flame, it is anticipated thatdifferential diffusion effects are important within the complete length of the domain.

4.1. Distribution of flame composition and extinction

Figure 4 shows mass fractions, at the same time and location, of methane, oxygen,water, carbon monoxide, molecular hydrogen and hydrogen radical. Plots of carbon

246 C. Pantano

Flow

Period

Figure 5. Three-dimensional rendering of hydrogen radical mass fractionat an instant in time.

dioxide are not shown because they are very similar to those of water in this flame.The extinguished regions are clearly seen in figure 4(f ), where gaps in the hydrogenradical mass fraction reflect extinction. In our flame, hydrogen radical is the onlyradical of the flame.

Availability of the hydrogen radical field facilitates geometrical studies of the flamebecause, in our case, although a flame is more complicated than just a one-fieldquantity, it can be identified well with the flame. Figure 5 shows the hydrogen radicalmass fraction field at one instant in time with the observer at two different angles. Thisfigure is a three-dimensional volume rendering of the field, where the magnitude of themass fraction determines the opacity of the zones. A nonlinear mapping was used tohighlight the regions of very large radical concentration. The images were generated byvolume rendering graphic cards from the Center for Advanced Computing Research(CACR) at Caltech. The view angle of figure 5(a) corresponds approximately to thatof an observer placed in the general direction of the jet and is slightly above the exitplane. Figure 5(b) is a frontal view of the flame, the jet is coming towards the observer.In both figures, the hydrogen radical mass fraction shows the large extinguished regionin the centre of the flame and the formation of multiple holes of varying geometries.The regions of high mass fraction, shown in dark contrast, were typically seen aroundvividly burning flame edges, as in closing holes, and were absent in regions that wereundergoing extinction. The peak value of hydrogen radical mass fraction aroundclosing holes or advancing edges was typically four to five times higher than the valuesobserved in the other regions of the flame. Note that some large holes in the flame,primarily in the upper region, appear unclosed. This is because the flow is periodic inthe spanwise direction and, at this time, the apparently missing section of the hole lieson the other side of the computational domain. Visualizations of the heat release ratefrom the DNS of Mahalingam et al. (1995) and Bedat et al. (1999) also indicate the


8(a) (b)

6

4

2

0 0.2 0.4 0.6 0.8 1.0Z Z

T

4× 10–4

3

2

1

0 0.2 0.4 0.6 0.8 1.0

YH

Figure 6. Scatter plot of (a) temperature and (b) hydrogen radical mass fraction versusmixture fraction at x/H = 11.

presence of holes in a diffusion flame submerged in a homogeneous turbulence fieldusing a synthetic chemical mechanism. The differences between our study and that ofBedat et al. (1999) reside in the chemical mechanism and the flow configuration.

Figure 6 shows scatter plots, collected in time, of temperature and hydrogen radicalmass fraction versus mixture fraction at the fixed streamwise location of x/H = 11. Ascan be seen in figure 6(a), temperature dependence on mixture fraction shows largescatter between the equilibrium values (upper envelope) and the frozen flow value(lower envelope). Figure 6(b) is complementary to the temperature and shows largescatter on hydrogen radical mass fraction owing to the constant extinction/reignitionof the flame.

4.2. Pilot stabilization mechanism

The role of the pilot in the stabilization of the flame and the choice of parametersare discussed next. Pilots are typically used in diffusion flame jets at sufficiently largeReynolds numbers because the rate of scalar dissipation is maximum close to theburner exit plane and decreases with increasing distance downstream. In these flows,the flame is unable to maintain the high temperatures required for combustionclose to the burner exit without an additional heat source. Furthermore, for coldreactants (our case) typical hydrocarbon flames are unable to autoignite downstreamand we are left with a turbulent non-reacting jet. An approach used frequently inexperimental investigations is to surround the main jet by a slower hot flow resultingfrom secondary combustion of another fuel. This low-momentum flow, low densityand high temperature, is typically composed of a mixture of reaction products andoxygen (Barlow & Frank 1998) and it is called the pilot flame. When the main jetcomes into contact with this hot flow, a flame is established and burns independentlyof the rate of scalar dissipation because the high temperature that control the typicalArrhenius nonlinear reaction rates is maintained externally. The temperature of thehot products decreases gradually with increasing distance until the flame starts to burnat a rate controlled by the scalar dissipation (that now is only a fraction of its valueat the burner exit). This technique is well suited to flows at high Reynolds numberswhere extinction can take place further downstream of the burner exit through thelocally large rate of strain caused by the intermittent nature of turbulence. In thecontext of numerical simulation, the difficulty with this technique is that the effect

248 C. Pantano

–10 –8 –6 –4 –2

ln(χ)

0

0.2

0.4

0.6

0.8

Pro

babi

lity

den

sity

fun

ctio

n

Figure 7. Conditional scalar dissipation p.d.f. at Z = Zs in the pilot controlled region,x/H < 2. The thick vertical line denotes the quenching limit from the flamelet equation.

of the pilot is felt for a relatively large distance from the burner exit and directsimulations require prohibitively large domains.

For lower-Reynolds-number flows, it is possible to create a pilot that consists ofan ignition source located at the burner exit plane (Yamashita et al. 1996). Thisis modelled in our simulation through the boundary conditions detailed in § 3.3,by maintaining the temperature high and the scalar dissipation low around the pilotinflow region. With this technique, it is found that for given Reynolds and Damkohlernumbers there is a parameter window, pilot thickness and velocity, over which theflame can be stabilized and, at the same time, the influence of the pilot can be limitedto a short region downstream of the burner exit. In this transitioning regime of pilotedflames, we can reliably stabilize the flame if the inflow parameters are well controlled;the case of a numerical simulation. In the present study, the parameters of the pilotwere determined by performing a number of two-dimensional simulations until thejet evolution was satisfactory.

Quantitative evidence that the pilot used in this study releases only small amountsof energy is evident in figure 5. In that figure, it can be seen that a flame hole formsin the lower pilot flame of the jet, close to the inflow. Several holes are formed duringthe course of the simulation, but they are unable to tear apart the pilot flames. Thepresence of these flame holes is evidence that the amount of energy introduced bythe pilot is limited. The pilot flame is only broken apart when it encounters the firststrong vortical structures in the centre of the domain. The strength of the pilot at theinflow can be inferred from figure 7, where the scalar dissipation p.d.f. conditioned onthe stoichiometric surface in the region x/H < 2 is shown. Here, the scalar dissipationis maintained mostly below the extinction limit, shown as a thick vertical line, owingto the quasi-laminar state of the flow. The higher temperature and associated higherviscosity and diffusivity generated by the pilot flame renders the local flow morestable and that helps to limit the magnitude of the scalar gradients. It appears thatthis is the mechanism by which the pilot flame is sustained.

The stabilization of the main flame further downstream, the region of interest inthis study, is caused by the intermittently broken pilot flames. When the pilot flameis broken by the first large-scale vortex, occurring in the region 3 <x/H < 5, flamesegments are convected downstream until they aggregate to the main flame. This


(a)

(b)

Figure 8. Three-dimensional renderings of hydrogen radical mass fraction showing the pilotflame annexation. (a) Top and (b) bottom pilot flame annexation events.

happens in the simulation at more or less regular intervals and helps to stabilize themain flame. Figure 8 shows a hydrogen radical mass fraction rendering at instants atwhich pilot flame segments are observed in the middle of the domain. These flamesegments travel downstream and contribute to the global stability of the main flameby joining it and increasing its flame surface. This stabilization mechanism is due todirect flame annexation.

5. Statistical characterization of the flowIn this section, we provide some statistical characterization of the flow. We define

the characteristic flow transient time as tL = Lx/Uj . Experience with this and other

250 C. Pantano

0 10 15x/H

1

2

3

4

5

2δ05

2δZ

5

Figure 9. Average velocity and mixture fraction jet thickness dependence onstreamwise coordinate.

kinds of turbulent flows show that, in order to achieve well-converged first-orderstatistics one must sample the flow for approximately 10 tL (Stanley et al. 2002;Jimenez 2003). In our case, such a simulation would have required approximately twoyears of computational time, with our present resources, and it is simply unattainableat this time. For this reason, we were able to run the simulation for approximately2 tL and we do not assert that statistical averages of all quantities extracted from thedatabase are well converged, but we believe that the present results are sufficient forthe study of extinction dynamics. This is a phenomenon occurring at the smallestscales of the flow, and averaging on time and across space on these regions givesreasonable statistical information. The parameters of the simulation were selected inorder to obtain sufficient extinction for averaging across these structures. Given theselimitations and in order to characterize the flow to some extent, we provide in thissection some Favre-averaged mean quantities.

Mean velocity, scalars, turbulence kinetic energy and scalar variances at differentsections across the flow are presented next. The average value of an arbitrary functionψ is computed as a simultaneous temporal and spanwise direction means,

ψ(x, y) =1

NT Nz

NT∑j=1

Nz∑i=1

ψ(x, y, zi, tj ), (5.1)

where NT is the number of time steps over which the average is computed. Favreaverages, ψ = ρψ/ρ, and Favre fluctuations, ψ ′′ = ψ − ψ , are defined in the usualway. All average values reported in this section were computed from t = 5 to t = 30.The values at the beginning of the simulation are not used in the calculation ofthe averages because the flamelet solution is still adapting to the flow conditions.Statistics concerning extinction dynamics are postponed to the following section.

Figure 9 shows the average jet width as a function of the streamwise coordinate.Two measurements of the jet width are provided in this figure: δ05 is the width based inthe 50% mean streamwise velocity profile and δZ is the width based on the 50% meanmixture fraction profile. Both thickness measurements are similar up to x/H ∼ 7, butδZ becomes larger than δ05 beyond this point. These results are consistent with thoseof Stanley et al. (2002) for non-heated planar jets. They also observe a transition in


–3 –2 –1 0y/δ05

–0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

x/H = 3.28.613.5

1 2 3

Figure 10. Average velocity at different streamwise positions.

–3 –2 –1 0 1y/δZ

0

0.2

0.4

0.6

0.8

1.0

1.2

x/H = 3.28.613.5

2 3

Figure 11. Average mixture fraction at different streamwise positions.

growth rate around x/H ∼ 7 and a faster rate of mixing for the scalar profile beyondthis point when compared to the rate of spread of the velocity profile.

Average streamwise velocity and mixture fraction profiles are shown in figures 10and 11 at three stations, x/H = 3.2, 8.6 and 13.5. It is observed that the mixturefraction profiles diffuse faster than the velocity profiles, similar to the results found inDNS of nonheated planar jets. Figure 12 shows average temperature profiles at thesame stations. It is seen that the average temperature is large, close to the adiabaticflame temperature, in the piloted region of the flame close to the inflow. In theintermediate station, the mean temperature is low owing to the large amount ofextinction that takes place at and around x/H =8.6. At the later station, x/H = 13.5,the temperature is higher than that of the preceding station owing to the lower levelof extinction in this region.

Figure 13 and 14 show average turbulence kinetic energy, k, and mixture fraction

variance, Z′′2, respectively. It is observed that the level of turbulence fluctuationsincreases with increasing distance from the inflow plane. The mixture fractionfluctuations also increase with increasing distance from the inflow plane, but it startsto decrease at the last station. This is consistent with the larger rate of mixing observed

252 C. Pantano

–3 –2 –1 0y/δ05

1

2

3

4

5

6x/H = 3.2

8.613.5

1 2 3

Figure 12. Average temperature at different streamwise positions.

–3 –2 –1 0y/δ05

0

0.02

0.04

0.06

0.08

x/H = 3.28.613.5

1 2 3

Figure 13. Average turbulence kinetic energy at different streamwise positions.

–3 –2 –1 0y/δZ

0

0.02

0.04

0.06

0.08

0.10

0.12

x/H = 3.28.613.5

1 2 3

Figure 14. Average mixture fraction variance at different streamwise positions.


–3 –2 –1 0 3y/δ05

0

0.1

0.2

0.3 0

5 × 10–4

1 × 10–3

0

0.1

0.2

0.3 0

5 × 10–4

1 × 10–3

1 2 –3 –2 –1 0 3y/δ05

1 2

(a) (b)

Figure 15. Average composition mass fractions at (a) x/H = 3.2 and (b) x/H = 13.5.�, CH4; �, O2; �, H2O; �, CO2; �, CO; �, H2 and �, H.

–2 –1 0 2y/δ05

–0.0015

–0.0010

–0.0005

0

x/H = 3.28.613.5

–5

–3

–1

1

3

5× 10–5

x/H = 3.28.613.5

1 –2 –1 0 2y/δ05

1

(a) (b)

Figure 16. Average reaction rates of (a) CH4 and (b) H at different values of x/H .

in figure 9. The results of figure 13 and the turbulence dissipation, ε, that are notshown here, give an estimate of the turbulence Reynolds number, Ret = urmsl/ν wherel = u3

rms/ε and u2rms =2k/3. These quantities varied across the jet width and reached

a peak, in most cases, around the centre of the jet. Our statistics give a peak Ret

around 300 at x/H = 3.2 and x/H = 13.5 and a value around 500 for x/H = 8.6. Thereason the Reynolds number is large around the centre of the computational domainis probably caused by two effects. First, in this region the flame is predominantlyextinguished with low temperature and high density. Secondly, the strong dependenceof the viscosity with temperature creates a low-viscosity region at the centre ofour domain. These effects contribute to a larger value of the turbulence Reynoldsnumber.

Figure 15 shows average composition mass fractions (a) close to the inflow planeand (b) close to the outflow plane. Figure 15(a) shows that the flame is almost planarin the piloted region close to the inflow, with negligible amounts of products at thecore of the jet. On the other hand, figure 15(b) shows that substantial mixing hastaken place at this station. Furthermore, figure 16 shows the average reaction ratesof methane and hydrogen radical at different distances from the inflow plane. It isseen that the methane reaction rate is relatively compact close to the inflow plane,

254 C. Pantano

it becomes very small in the central region where large-scale extinction is presentand becomes broader further downstream where the flame is burning again and it isbeing stirred by turbulence. On the other hand, the reaction rate of hydrogen radicalbehaves differently. In fact, while the reaction rate average is relatively compact closeto the inflow plane, it becomes almost negligible far downstream. This behaviour, atfirst puzzling, is a consequence of the dependence of the hydrogen radical reaction rateon the spatial coordinates. The reaction rate of hydrogen radical has both positiveand negative parts. When averaging in time is applied at a specified location in space,these positive and negative contributions tend to cancel each other out. This results invery small values of the average reaction rate for the radical. This is not observed formajor species because the reaction rates are predominantly positive for products andnegative for reactants. Although our results are not conclusive owing to our limitedsampling, it is clear that if these observations results are correct, the implications formodelling purposes are important.

There have been some attempts to model chemically reacting flows by solvingaveraged transport equations for all the chemical species. In this approach, the averagereaction-rate terms must be modelled somehow. This methodology is commonlyreferred to as the direct-closure approach and constitutes the turbulent-combustion-closure problem, properly speaking. Our statistics indicate that attempts to modelaverage radical reaction rates are futile, because the positive (production) and negative(consumption) parts of typical radical reaction rate functions combined with thespatial and temporal variability of turbulent flows produce average rates that are notrepresentative of their instantaneous values. This is important because it implies thatone cannot model average radical reaction rates from the knowledge of the ratesprovided by the detailed chemical mechanism.

6. Flame-edge statisticsA turbulent flame is a complex geometrical object. We can think of it as a coupled

multiscalar manifold with changing topology. The coupling originates in the reactionrate terms that represent chemical conversion and affect in a direct way thetemperature, density and mean molecular weight. These direct variations, primarilyin density, but also in molecular properties through variations of the temperature,induce indirect changes in the velocity field. These velocity variations can, in turn,affect the rates at which reactants and products are brought together and removedfrom the flame, respectively. This coupling is generally referred to in the literature asthe turbulence–chemistry interaction. Depending on the combustion regime, severaltheoretical abstractions have been used to understand this coupling. For example, ithas been proposed (Williams 1975) that, under the appropriate conditions of largeDamkohler numbers, the flame becomes very thin. In this case, the geometry of theflame is relatively well defined by a surface, the so-called flame sheet.

The results of the present study support, to some extent, the idea that even fordiffusion flames under relatively large rates of strain, the flame remains quite thin(see figure 4f ). We are certainly assuming that the mass fraction of hydrogen radicalcan be realistically used as a marker of the flame. In our case, there are only a fewmeaningful fields that could be used to identify the flame. It is common to use theheat release rate to associate the regions where the flame burns vigorously with theflame location, see Im & Chen (2001) for an example. In our case, a quick reviewof the algebraic form of the reaction rates of the reduced mechanism used in thisstudy reveals that all these rates are proportional to the concentration of hydrogen


radical. Some differences between the hydrogen radical field and the heat release rateare bound to exist, specially during short transients. These transients are likely toappear at the instant a hole is formed and also when a hole collapses. In the firstcase, as soon as these extinguished regions grow, the hydrogen radical will diffuse andaccommodate around the flame edges and disappear from the completely extinguishedregion. In the second case, it has been shown that the collapse of flame holes is a veryfast process (Buckmaster & Jackson 2000; Pantano & Pullin 2003) and it should nothave a statistical impact on the results. Thus, we expect that, apart from these veryshort transients, the hydrogen radical mass fraction is a reasonable indicator of ourflame location. We would like to stress that this choice is by no means unique andinvolves some degree of uncertainty.

Two statistical quantities are investigated in this section. First, we determine thetotal flame area evolution with time within the computational domain. Secondly,once the flame edges are identified, the joint flame-edge velocity-scalar dissipationprobability distribution (p.d.f.) is recovered.

6.1. Flame identification

We assume that the external structure of the flame is approximately defined by themixture fraction field, such that the flame lies around a mixture fraction surface,Z(x1, x2, x3, t) =Zo, where Zo is close to Zs . This definition is appropriate far fromextinction and when the Lewis number of all species is one (Williams 1985). Inour case, detailed visualization of the flame structure shows that this definition isstill appropriate, even though the Lewis number is not unity (see beginning of § 4),provided the extinguished surface regions are removed. This is simply accomplishedby recognizing that the mass fraction of hydrogen radical is negligible in those regions.We define three subspaces, according to

S(t) = {(x1, x2, x3) ∈ R3 : Z(x1, x2, x3, t) = Zo}, (6.1)

F(t) = {(x1, x2, x3) ∈ R3 : Z(x1, x2, x3, t) = Zo, YH(x1, x2, x3, t) > Ys}, (6.2)

E(t) = {(x1, x2, x3) ∈ R3 : Z(x1, x2, x3, t) = Zo, YH(x1, x2, x3, t) = Yo}, (6.3)

where S(t) is the two-dimensional manifold defining the stoichiometric surface, F(t)is the two-dimensional manifold defining the flame and E(t) is the one-dimensionalmanifold defining the flame edges. F(t) is a three-dimensional surface with holes andE(t) is composed of multiple open and closed loops. The parameter Yo is a threshold ofthe hydrogen radical mass fraction and it has been chosen after extensive visualizationof the flame edges and is typically 5 to 10% of the peak hydrogen radical mass fractionon the flame, in our case Yo = 3 × 10−5. This value has proved to give very reliableresults regarding identification of the flame and flame edges, except at those locationand instants when a hole is created. These occurrences cannot be detected accuratelywith the present algorithm, but they do not represent a large fraction of the edges interms of sampling numbers and they are removed from the statistics. It was foundthat using the hydrogen radical field produced more reliable identification of theflame edges than when the heat release rate was used. This seems to be due to thecomplex structure of the heat release rate around edges (Ruetsch et al. 1995). Finally,a comparison of the flame surface determination based on the Bilger, Starner & Kee(1990) mixture fraction variable was carried out and no meaningful differences wereobserved. Both mixture fractions give equally good flame surface approximations.This is probably due to the manner in which the flow is initialized, since both theinitial flow and the forcing are mapped to Z. Using the previous definitions, we

256 C. Pantano

y

x

z

y

x

z

y

x

z

y

x

z

02

4 02

46

810

1214

z

x 02

4 02

46

810

1214

z

x

02

4 02

46

810

1214

z

x02

4 02

46

810

1214

z

x

(a) (b)

(c)(d)

Figure 17. Flame surface and edge geometry at four times during the simulation.

developed an algorithm that extracts F(t) and E(t). The algorithm is described indetail in Pantano & Lombeyda (2003). Further details regarding the effect of varyingisosurfaces are reported in the Appendix. The following flame surface visualizationand statistics correspond to case (ii) in the Appendix.

Figure 17 shows the flame surface F(t) and the flame edges E(t), thick black lines,at several instants in time during the simulation. The jet runs from left to rightand the two flames at the opposite sides of the jet are visible. Since the domain isperiodic in the spanwise direction, two copies of the flame are shown as a visual aideto help in the interpretation of the surface geometry. The region to the left showsthe well-defined pilot stabilized region. Figure 17(a) corresponds to time t = 0.3tL,after the initial adjustment of the flamelet profiles. Figures 17(b)–17(d) correspond tot = 0.8tL, 1.3tL and 1.8tL, respectively.

In order to make more quantitative measurements, we define the area operator by

A(X) =

∫X

dX, (6.4)

where X is the surface manifold coordinates in the three-dimensional space and dXdenotes the area differential. The postprocessing algorithm was used to extract aburning index defined as the ratio of flame area to stoichiometric surface area,


0 10 20 30t

0.3

0.4

0.5

0.6

0.7

0.8

A(�)–—

A(�)

Figure 18. Burning index evolution with time.

given by

r =A(F)

A(S). (6.5)

Figure 18 shows the burning index, r , as a function of time. The values of r duringthe first instants of the simulation are not reported because the algorithm was unableto reliably predict the burning flame surface. This is due to the strong transient effectsintroduced by the relaxation of the initial flamelet profiles to the correct values.Figure 18 shows that, initially, r has a relatively high value, owing to the artificialinitial condition, and decreases strongly. Then, r stabilizes somewhat, from t = 5 to 20.This relatively calm period is followed by a further decrease of r . The origin of thisdecrease is partially due to the presence of a large-scale organized vortex that wrapsthe flame and convects a large portion of the burning flame outside the domain, andmakes the value of r temporarily low. Availability of more powerful computationalresources in the future will certainly allow larger computational domains and times toextend the current results further. Nevertheless, the central period of our simulation,where r is relatively uniform, can confidently be used to extract meaningful statisticaldata about the flame edges since the total flame area does not change much.

6.2. Flame-edge velocity statistics

At this point, we will use the previously detected flame edges to extract statisticsabout the flame-edge velocity. We start by defining the unitary normal directions tothe mixture fraction and hydrogen radical mass fractions isosurfaces

nZ =∇Z

|∇Z| , (6.6)

nH =∇YH

|∇YH| , (6.7)

258 C. Pantano

VH

V β VZ

θ

B

A

Hydrogen isosurface

Mixture fraction isosurfaceα

Flame

Figure 19. Geometrical determination of the edge velocity.

respectively. The normal velocity of the isosurfaces can be determined from knowledgeof the transport equations of the respective fields and they are given by

VZ = u · nZ − ∇ · (ρDZ∇Z)

ρ|∇Z| , (6.8)

VH = u · nH − ∇ · (ρDH∇YH) + ωH

ρ|∇YH| , (6.9)

for mixture fraction and hydrogen radical mass fraction, respectively. The diffusivitiesof mixture fraction and hydrogen radical are given by DZ = δ∗(T )/(Re Sc) andDH = δ∗(T )/(Re ScH ), respectively. Notice that both the flow velocity and the normaldirection intervene in the definitions. Moreover, it is well known that only the normalvelocity of the isosurfaces can be uniquely defined. It is not uncommon to com-plement the normal velocity with the component of the flow velocity tangential to thenormal. In doing so, a full three-dimensional velocity vector can be associated at eachisosurface point (Gibson 1968). Here, we do not require this extension of the isosurfacevelocity because the normal velocity suffices. Figure 19 shows a generic geometricdisposition of the isosurfaces, the normal velocities and the edge displacement. Notethat this figure only shows one of the possible cases, that in which the sign of VH ispositive, that is, the isosurface velocity is in the same direction as the normal. Thecase in which VH is opposite to nH (VH < 0) is also possible and leads to a similargeometrical treatment with some angles that are complements of those shown infigure 19. For compactness, we show the derivation for the case where VH > 0 shownin figure 19. The angles α and β denote the angles of the normals to the edge velocityvector, determined in the figure by the vector that goes from point A to B. Theseangles are related to the angle between the normals, θ , by

θ = α + β, (6.10)

where

cos θ = nZ · nH. (6.11)


These angles can be used to relate the velocity of point A, V , to those of theisosurfaces, through

V cos α = VH, (6.12)

V cos β = VZ. (6.13)

Equations (6.10)–(6.13) can be manipulated to obtain explicit relations for V, α andβ in terms of θ , VH and VZ . At this point, we define the relative velocity of the flameedge, Vo, as the projection of V on the stoichiometric surface with the conventionthat positive Vo implies edges travelling in the direction of the extinguished regionand it is defined by

Vo = −V cos(

12π − β

). (6.14)

In order to determine what is called the flame-edge speed, similar to the definitionused by Ruetsch et al. (1995), in our approach we must take into account the velocityof the flow incoming towards the edge. This velocity is given with our convention by

um = u · m, (6.15)

where m is the unitary tangent vector at the edge,

m = nZ × (nH × nZ).

Note that um is positive in the opposite direction to Vo, that is, in the direction of theburning region. With all these definitions at hand, the flame-edge velocity is given by

Ve = Vo + um. (6.16)

Other works (Ruetsch et al. 1995; Im & Chen 1999) have used a procedure that isvery similar to that introduced above, equations (6.6)–(6.16), to determine the velocityof intersecting isolines. In our case, apart from the fact that the edges are three-dimensional, our choice of scalar fields is different and we project the velocity back tothe stoichiometric surface to satisfy the premise that the edges are supposed to moveon this surface.

Equations (6.6)–(6.16) and the scalar dissipation χ were computed at each ofthe flame-edge locations extracted by the edge identification algorithm describedpreviously. The data were accumulated from approximately 400 times to constructa joint p.d.f. Moreover, only the downstream half of the computational domain wasused to compute joint statistics. This was done to avoid mixing the statistics from thepilot region, where edge dynamics may be different, with those of the region of interestdownstream. Figure 20 shows a contourmap of the joint flame-edge velocity-scalardissipation probability density function. The horizontal axis shows the flame-edgevelocity in linear coordinates and the vertical axis shows the natural logarithm of thescalar dissipation at the edge. The equally spaced joint p.d.f. values are represented bydifferent tonalities of grey. Although some statistical scatter is present, the joint p.d.f.is reasonably converged. It can be seen that the joint p.d.f. has a characteristic shapethat resembles the form of the dependence of the edge velocity on scalar dissipationof laminar studies (Daou & Linan 1998), but in the present study, unstationary effectsare clearly visible. Note that the joint p.d.f. is effectively broad and we expect thatthis is a manifestation of the randomness of the flow. Moreover, it is seen that asthe scalar dissipation becomes large, the only probable values of the joint p.d.f. arethose for which the edge velocity, Ve, is negative, that is, receding edges or expandingholes. The quenching value of the scalar dissipation determined previously from theflamelet equation is shown here as a horizontal thick line for reference purposes. It

260 C. Pantano

Ve

ln(χ

)

–0.5 0 0.5–8

–7

–6

–5

–4

–3

Figure 20. Joint flame-edge velocity-scalar dissipation p.d.f. The thick horizontal line re-presents quenching value of scalar dissipation, χq , and the vertical line represents stabilizationflame-edge speed.

is expected that the flame ceases to exist for values of the scalar dissipation aroundthe quenching value. In this figure, some edges propagating with negative velocitiesare encountered in the regions where χ is somewhat larger than the laminar quenchingvalue, χq . This has been observed in the past by Mahalingam et al. (1995) in DNS ofturbulent non-premixed combustion. They also identify the fact that the flow boundaryconditions used in laminar calculations can influence the precise numerical value ofthe extinction limit. These boundary conditions cannot capture all unstationary andthree-dimensional effects. For these reasons, we do not expect that the quenchingvalue, χq , obtained from any specific one-dimensional configuration of the flame willgive quantitatively accurate values in three-dimensional flows, though, typically theagreement is very good.

On the other limit of χ , as the scalar dissipation becomes small, the joint p.d.f.is non-negligible towards positive Ve. Based on previous works of two-dimensionalsimulation of edge flames, it is expected that this vertical asymptote should be centredaround the stabilization edge speed. This speed is estimated here, following Ruetschet al. (1995), as the product of the laminar premixed speed at the stoichiometricconditions, SL,st , multiplied by the square root of the density ratio of the frozen flow,ρf , to that of the diffusion flame, ρb. The value we estimate is SL,st

√(ρf /ρb) = 0.058

and is shown in figure 20 as a vertical thick line. It can be seen that the peak of thejoint p.d.f. is centred around this value in this region.

6.3. Heat release rate statistics

Among the multiple statistics that can be investigated in turbulent non-premixedcombustion with extinction, the correlation between heat release rate and scalardissipation has been the subject of increased attention. This correlation has beeninvestigated in the past using DNS by Mahalingam et al. (1995) for one- and two-step chemistry and by Swaminathan et al. (1996) using single-step chemistry. Theyobserve, in accordance with laminar theory (Peters 1984), that the heat release rateincreases with increasing scalar dissipation. Figure 21 shows the conditional joint


ln(χ)

ln(ω

T)

–8 –6 –4–5

–4

–3

–2

–1

0

Figure 21. Conditional joint p.d.f. of heat release reaction rate and scalar dissipationat the stoichiometric mixture fraction, Zs .

p.d.f. of heat release rate, ωT = −Da∑N

i=1 hiωi , and scalar dissipation collected in anarrow band of thickness 0.02 in mixture fraction space around the stoichiometricsurface, Zs . This joint p.d.f. was compiled at the same times as the edge velocity–scalardissipation joint p.d.f. and does not include the pilot flame, x/H > 6. The horizontalaxis denotes the natural logarithm of the scalar dissipation and the vertical axis isthe natural logarithm of the heat release rate. The isocontour levels are denoted intonalities of grey; where the graduation from dark to light grey represent higher tolower uniform isolevels, respectively. The thick line on the same figure denotes thepeak heat release rate as a function of scalar dissipation obtained from the flameletequation, (3.42). The power law for this flamelet solution is approximately 0.685. Incontrast, the conditional joint p.d.f. evolves approximately aligned with the directionof unity slope. These differences, apart from the broad character of the distributioncaused by unstationary effects, probably originates from the contributions of thedifferent modes of combustion in the flow; ranging from the burning flame edgesto the extinction events. Since, in our regime of extinction, the number of flameedges/extinction events is important, their contribution to the statistics of the heatrelease rate are obviously present. It can be seen that at higher values of the scalardissipation, the alignment with the flamelet solution is good. On the other hand,at lower values of the scalar dissipation, this alignment is less pronounced and theflamelet solution appears shifted upwards with respect to the joint p.d.f. In this region,we expect to encounter the contributions from the flame holes that are closing withpositive flame-edge velocities; compare with figure 20. Here, the prevalent burningmode of the edge flames will contribute to the observed deviation of the p.d.f. statisticsfrom the flamelet solution. The improved agreement of the flamelet solution for highscalar dissipation rates has been observed by Mell et al. (1994) in DNS of constantdensity reacting flows. They report that this is associated with an increase in theone-dimensionality of the reaction zone.

7. DiscussionPrevious two-dimensional numerical works have investigated the mechanisms and

parameters that determine the flame-edge velocity. In the case of stationary edges, this

262 C. Pantano

velocity is well defined and Daou & Linan (1998) give a detailed account for a one-step chemistry model with constant density. Other numerical works of unstationaryflame edges subjected to the strain field of a vortex include Favier & Vervisch (1998),Im & Chen (1999) and Im & Chen (2001). In this case, a flame edge is approximatelyaligned with the centre of a vortex system and the edge propagates upstream ordownstream depending on the relative strength of the vortex with respect to thechemistry. Boulanger & Vervisch (2002) studied the expected value of the Damkohlernumber at the tip of an edge flame, taking into account the ratio of the diffusivefluxes normal to the non-premixed flame to the premixed fluxes at the tip of theflame. They find that this ratio plays a role in the prediction of the flame-edge speed.In general, all studies agree that some measure of the mixture fraction gradient at theflame edge controls the propagation of the structure and that chemistry details cannotbe neglected. In the present simulation, the analysis of the statistics suggest that, inthree-dimensional flame edges, the scalar dissipation can be used to parameterize theedge speed. Unstationary effects are important and are reflected by a somewhat broaddistribution function. These results can be exploited in advanced modelling of flameextinction/reignition through flame-edge propagation for turbulent combustion in twoways. First, we can neglect unstationary effects and assume that the edge-flame speedis determined uniquely by the local instantaneous value of the scalar dissipation. Thisfunction can be obtained from a two-dimensional boundary-value problem involvingthe chemistry, transport and heat release details for a certain canonical flow config-uration, as in the works reported by Ruetsch et al. (1995) and Daou & Linan (1998),as a function of the scalar rate of dissipation. This information can then be used toconstruct a triple-flamelet closure, first suggested by Dold, Hartley & Green (1991), bytaking into account the statistics of the scalar dissipation at the stoichiometric surface.An example of this approach has been attempted by Pantano & Pullin (2004) for smallflame holes. In the second approach, it may be possible to account for the unstationaryeffects on the flame-edge speed by either solving the corresponding joint-p.d.f.transport equation for flame-edge speed-scalar dissipation or adapting some of theideas of the second-order conditional moment closure (CMC) method of Klimenko &Bilger (1999). In this latter case, additional correlations of the flame-edge speed due tounstationary effects caused by the statistics of the scalar dissipation could be retained.

Finally, of the two mechanisms that are thought to be primarily involved inreignition dynamics, flame-edge propagation and ignition through heat conductionfrom nearby hot products, only the former is discussed here. It can be seen in figure 20that the simulation parameters were chosen appropriately to cover both expandingand collapsing holes. This is shown by the occurrence of both positive and negativevalues of the flame-edge velocity in the joint p.d.f. In the second mechanism, pocketsof burned hot gases are convected and come into close proximity of fuel–oxidizermixtures from the extinguished region leading to reignition. This mechanism cannotbe captured in the present simulation. The reduced chemical mechanism used in thisstudy is derived assuming that certain radical species are in quasi-steady state andthat some reactions are in partial equilibrium. The species and reactions that areselected are appropriate for the burning regime but they are inappropriate for thequenched state (Peters 1985). In fact, Peters (1985) recommends that an alternativereduced mechanism should be derived if one is interested in the ignition phenomena.We can go back to the algebraic relations describing the reaction rates in § 3.2 andobserve that the chemistry is controlled by the hydrogen radical. This radical can onlyexist around the flame and it is not encountered, in our flame, at any concentration inthe extinguished regions. Thus, lack of hydrogen radicals make it impossible for this


flame to reignite, no matter how much time is given to the system. This limitation inour chemistry mechanism is recognized in the present study.

8. ConclusionWe report results of a direct numerical simulation of a turbulent non-premixed

methane–air planar jet using a four-step reduced mechanism. Owing to the large costof the simulation, the computational domain is restricted to the near-field region ofthe flow. The parameters of the simulation were selected to exhibit a non-negligibledegree of extinction in order to study the dynamics of diffusion flame edges. Thefour-step reduced mechanism is the simplest reduced mechanism in the hierarchyof reduced mechanisms that includes radicals. In our case, the only radical that iscomputed along with the flow and stable species is the hydrogen radical.

Turbulence statistics were collected in time from the simulation database. It wasfound that average radical reaction rates in the more turbulent regions of the floware negligible in comparison with their instantaneous contributions. This is becausethe radical reaction rates alternate signs across the flame and the fluctuating natureof the turbulent flow averages out most of the contributions to the average.

A feature identification algorithm was developed to extract flame-edge statisticsfrom the simulation database. The flame is assumed to be spatially defined by thehydrogen radical mass fraction field. The flame edges are identified as the curves inspace where a low-value isosurface of hydrogen radical mass fraction and the stoichio-metric mixture fraction isosurface intersect. This geometric reduction approach wasshown to give very reliable locations of the flame edges and holes. Knowledge of thetransport equations of the scalar fields was then used to extract joint statistics of theflame-edge velocity and scalar dissipation (a local Damkohler number). It was foundthat while the peak of the joint p.d.f. of these two quantities bears some resemblanceto the theoretically laminar flame-edge velocity relationship on scalar dissipation,substantial widening of the joint p.d.f. exists. This is presumably due to unstationaryeffects.

This work was supported in part by the ASC program of the Department of Energyunder subcontract no. B341492 of DOE contract W-7405-ENG-48. The author wouldlike to thank S. Lombeyda of the Center for Advanced Computing Research atCaltech for generating the three-dimensional hydrogen radical mass fraction figures.The author would also like to thank Professor D. I. Pullin for innumerable discussionsand for reading and suggesting improvements to the manuscript. Additionally, theauthor would like to thank the referees for many comments.

Appendix. Edge-detection sensitivityThis Appendix addresses some of the heuristic details associated with the edge-

flame-detection algorithm. The current scheme is based on an extension of theprocedure outlined by Ruetsch et al. (1995) for two-dimensional flame edges. Insteadof intersection of isolines, we must consider the intersection of isosurfaces in three-dimensional flows. In our flame, these isosurfaces are obtained from the mixturefraction and hydrogen radical mass fraction fields through (6.3). In order to quantifythe impact of the isosurface values on the quality and uncertainty of the measurededge velocity we have conducted a limited parametric study by considering four setsof isosurface values. There is a limited range of mixture fraction values and hydrogen

264 C. Pantano

Case Zo Yo

(i) 0.2 6 × 10−5

(ii) 0.2 3 × 10−5

(iii) 0.1875 3 × 10−5

(iv) 0.16 3 × 10−5

Table 3. Threshold values of Zo and Yo for parametric study of the edge-detection algorithm.

35 5

5

7

7

9

A

B

4

5

56

67 7

C

(a) (b)

Figure 22. Close up of two typical edges and the associated hydrogen radical mass fraction(grey contourmap), stoichiometric line (thick dashed-double-dotted line), hydrogen radicalmass fraction threshold (thick line) and hydrogen radical reaction rate (thin dashed-dottedline); (a) desirable case and (b) less desirable case. Hydrogen radical reaction rate isocontoursare labelled with numbers from 1 to 12 and denoting the values −0.002, −0.0015, −0.001,−0.0005, −0.00025, 0.00025, 0.0005, 0.001, 0.0015, 0.002, 0.0025 and 0.003, respectively.

radical mass fractions that are able to detect the flame edges accurately for all times(in our simulation) owing to the unstationary nature of the flame. This range isgiven approximately by 0.16 <Zo <Zs = 0.2 and Yo < 6 × 10−5 in our case. Valuesoutside this range tend to produce fictitious flame edges or failed to detect the edgesaltogether. The four cases described in this Appendix are given in table 3.

To illustrate the typical cases that were encountered in the simulation database,figures 22(a) and 22(b) show two-dimensional cuts of two typical situations with thethresholds of case (ii). The colour and line scheme is the following: grey isocontoursdenote the intensity of hydrogen radical mass fraction (from black, highest value,to white, lowest), the thick continuous line denotes the isoline corresponding tothe thresholds of hydrogen radical for case (ii), the thick dashed-double-dotted linedenotes the stoichiometric line on this plane and finally the dashed-dotted line denotesisolines of the hydrogen radical reaction rate. Figure 22(a) is a representation of thevertical plane that runs through the centre of the small hole shown in the frontal viewof figure 5. Two edges are shown in this figure. The case shown in figure 22(a) is a desir-able case from the point of view of edge detection, because there is a very sharp changeof the hydrogen radical mass fraction, and the isosurfaces, mixture fraction and hy-drogen radical form nearly orthogonal angles. Figure 22(b) shows another case wherethe quality of the edge detection is less good, in the sense that the angle between thetwo surfaces at the intersection point is rather small. These two figures depict the twotypical situations that were observed in the simulation database. Figure 23(a) showsthe hydrogen radical mass fraction through the vertical plane passing through theisosurface intersection, points A and B, of figure 22(a) and that through the horizontalplane at the intersection, point C, in figure 22(b). The threshold value is shown as a


YH

0

0.0001

0.0002

0.0003

0.0004

0.0005

A

0

0.0001

0.0002

0.0003

0.0004

0.0005

CB

(a) (b)

Figure 23. Hydrogen radical mass fraction variation across edges; (a) desirable case,(b) less desirable case.

–1 0 1cos (θ)

0

1

2

3

4

Pro

babi

lity

den

sity

fun

ctio

n (i)(ii)(iii)(iv)

Figure 24. Probability density function of cos (θ ) for varying threshold parameters.

broken thick line. It can be seen that the threshold value gives a reasonable detectionfor the edges, while the threshold used in case (i), twice as high, is probably too high forthe edges B and C. Nevertheless, the quality of the edge location is reasonable in bothcases. Had we had chosen a higher threshold, we would have been penalized in thequality of the detection of cases like that shown in figure 23(b). Given these observa-tions, it was deemed appropriate to use Yo =3 × 10−5 for the extraction of the statistics.

A quantity that is useful in assessing the quality of the edge detection is thedistribution of angles between the mixture fraction isosurface and the hydrogenradical mass fraction isosurface, θ . Very small angles have large uncertainty becausethey correspond to surfaces that run almost parallel to each other. Figure 24 showsthe p.d.f. of cos (θ) for all cases in table 3. It is observed that the p.d.f. is mostlyconcentrated in the region cos (θ) < 0. Furthermore, depending on the thresholdvalues, some p.d.f.s have appreciabe probability values at cos (θ) = −1. The samplesthat contribute to the p.d.f. at this location correspond, in the database, to flame holeformation events. They represent highly transient processes that cannot be detectedeasily because the flame edges have not formed yet. The flame is transitioning fromits almost one-dimensional to a two-dimensional structure (in the plane of the edge).These events are associated with very large rates of scalar dissipation and extremelylarge edge velocities. Fortunately, they represent a small fraction of all the edgesin the simulation and after some experimentation it was observed that we could

266 C. Pantano

ln(χ

)

–0.5 0 0.5–8

–7

–6

–5

–4

–3

–0.5 0 0.5–8

–7

–6

–5

–4

–3

ln(χ

)

–0.5 0 0.5–8

–7

–6

–5

–4

–3

–0.5 0 0.5–8

–7

–6

–5

–4

–3

Ve Ve

(i) (ii)

(iii) (iv)

Figure 25. Joint p.d.f. of edge velocity-scalar dissipation for all four cases in table 3.

exclude most of these events by discarding the cases with |cos(θ)| > cos (θo). Wedetermined that θo =10◦ was a satisfactory choice that resulted in the exclusion ofapproximately 4% of the samples. As shown below, this procedure has negligibleeffect on the measured statistics and helps to produce smoother p.d.f.s by removingthe contribution from the uncertain isosurfaces intersections.

Finally, the joint p.d.f. of flame-edge velocity and scalar dissipation using thedifferent isosurface thresholds in table 3 are shown in figure 25. It is observed that allthresholds produce p.d.f.s that are qualitatively and quantitatively very similar; somesmall degree of statistical variability is unavoidable. Moreover, the fact that edgeswhere |cos(θ)| > cos (θo) have been excluded has a negligible impact on the statistics.This is the case here, because by altering the threshold value, Zo, we also change theangle θ of each detected edge. By spanning the range of Zo values, from 0.16 to 0.2,a good edge that is detected with a poor angle at a given threshold becomes welldetected with a different threshold. A good edge is one that is well defined and it is notpart of a hole formation event. Since the computed p.d.f.s do not change appreciablyfor different threshold values, we can conclude that our exclusion criteria based on θ

remove most of the contributions from the ill-defined hole formation events.

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